Solve the equation.cosec x = - √2
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We know that principal value branch of
![{cosec}^{ - 1} x \:\: is\:\: [-\frac{\pi}{2}, \frac{\pi}{2}] \\](https://tex.z-dn.net/?f=+%7Bcosec%7D%5E%7B+-+1%7D+x+%5C%3A%5C%3A+is%5C%3A%5C%3A+%5B-%5Cfrac%7B%5Cpi%7D%7B2%7D%2C+%5Cfrac%7B%5Cpi%7D%7B2%7D%5D+%5C%5C+ "{cosec}^{ - 1} x \:\: is\:\: [-\frac{\pi}{2}, \frac{\pi}{2}] \\")
-{0}
To solve the given equation we must keep consider that cosec and cosec inverse cancels each other only if x belongs to its principal value .
![cosec\: x = -\sqrt{2}\\ \\ x = {cosec}^{ - 1} ( -\sqrt{2} )\\ \\ we \: know \: that \: cosec \: \frac{\pi}{4} = \sqrt{2} \\ \\We\:\: know\:\:that\:\: \\ \\ {cosec}^{ - 1} ( - cosec\: x) = - {cosec}^{ - 1} ( cosec\: x) \\\\ So\\\\ x= {cosec}^{ - 1} (cosec ( -\frac{\pi}{4} )) \\ \\x=- {cosec}^{ - 1} (cosec ( \frac{\pi}{4} ))\\\\ x= - \frac{ \pi}{4} \: \: \: \: (x \: belongs \: to \: [- \frac{\pi}{2}, \frac{\pi}{2}])-(0)\\ \\ x = \frac{ - \pi}{4} \\](https://tex.z-dn.net/?f=+cosec%5C%3A+x+%3D+-%5Csqrt%7B2%7D%5C%5C+%5C%5C+x+%3D+%7Bcosec%7D%5E%7B+-+1%7D+%28+-%5Csqrt%7B2%7D+%29%5C%5C+%5C%5C+we+%5C%3A+know+%5C%3A+that+%5C%3A+cosec+%5C%3A+%5Cfrac%7B%5Cpi%7D%7B4%7D+%3D+%5Csqrt%7B2%7D+%5C%5C+%5C%5CWe%5C%3A%5C%3A+know%5C%3A%5C%3Athat%5C%3A%5C%3A+%5C%5C+%5C%5C+%7Bcosec%7D%5E%7B+-+1%7D+%28+-+cosec%5C%3A+x%29+%3D+-+%7Bcosec%7D%5E%7B+-+1%7D+%28+cosec%5C%3A+x%29+%5C%5C%5C%5C+So%5C%5C%5C%5C+x%3D+%7Bcosec%7D%5E%7B+-+1%7D+%28cosec+%28+-%5Cfrac%7B%5Cpi%7D%7B4%7D+%29%29+%5C%5C+%5C%5Cx%3D-+%7Bcosec%7D%5E%7B+-+1%7D+%28cosec+%28+%5Cfrac%7B%5Cpi%7D%7B4%7D+%29%29%5C%5C%5C%5C+x%3D+-+%5Cfrac%7B+%5Cpi%7D%7B4%7D+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%28x+%5C%3A+belongs+%5C%3A+to+%5C%3A+%5B-+%5Cfrac%7B%5Cpi%7D%7B2%7D%2C+%5Cfrac%7B%5Cpi%7D%7B2%7D%5D%29-%280%29%5C%5C+%5C%5C+x+%3D+%5Cfrac%7B+-+%5Cpi%7D%7B4%7D+%5C%5C+ "cosec\: x = -\sqrt{2}\\ \\ x = {cosec}^{ - 1} ( -\sqrt{2} )\\ \\ we \: know \: that \: cosec \: \frac{\pi}{4} = \sqrt{2} \\ \\We\:\: know\:\:that\:\: \\ \\ {cosec}^{ - 1} ( - cosec\: x) = - {cosec}^{ - 1} ( cosec\: x) \\\\ So\\\\ x= {cosec}^{ - 1} (cosec ( -\frac{\pi}{4} )) \\ \\x=- {cosec}^{ - 1} (cosec ( \frac{\pi}{4} ))\\\\ x= - \frac{ \pi}{4} \: \: \: \: (x \: belongs \: to \: [- \frac{\pi}{2}, \frac{\pi}{2}])-(0)\\ \\ x = \frac{ - \pi}{4} \\")
is the solution of the equation.
To solve the given equation we must keep consider that cosec and cosec inverse cancels each other only if x belongs to its principal value .
is the solution of the equation.
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